11.1 Solving Polynomial Systems

Gröbner bases are powerful tools for solving polynomial systems. They provide a canonical representation of polynomial ideals, making it easier to analyze and manipulate complex equations.

Using Gröbner bases, we can determine the number and nature of solutions to polynomial systems. This approach also helps us understand the geometry of solution sets and handle various types of systems, from those with no solutions to those with infinitely many.

Gröbner Bases and Polynomial Systems

Application of Gröbner bases

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• Gröbner bases
  • Special generating set for an ideal in a polynomial ring enables solving systems of polynomial equations
  • Computed using Buchberger's algorithm or variants (Faugère's F4 and F5 algorithms)
  • Provides a canonical representation of the ideal, making it easier to manipulate and analyze
• Solving polynomial systems using Gröbner bases
  1. Compute the Gröbner basis of the ideal generated by the polynomials (using a monomial ordering like lexicographic or graded reverse lexicographic)
  2. Use the elimination property of Gröbner bases to eliminate variables and obtain a triangular system
  3. Solve the resulting univariate polynomial equation (using techniques like factorization or numerical methods)
  4. Back-substitute the solutions to find values for the other variables, working backwards through the triangular system

Solution count from Gröbner bases

• Finite number of solutions
  • Gröbner basis contains a univariate polynomial in each variable, indicating a finite number of solutions
  • Number of solutions is the product of the degrees of these univariate polynomials (Bézout's theorem)
  • Example: if the Gröbner basis contains polynomials of degrees 2 and 3 in $x$ and $y$ respectively, there are 2 × 3 = 6 solutions
• Infinitely many solutions
  • Gröbner basis does not contain a univariate polynomial in each variable, indicating infinitely many solutions
  • System has infinitely many solutions, typically parametrized by one or more free variables
  • Example: if the Gröbner basis contains only a polynomial in $x$, the solutions are parametrized by the free variable $y$
• Hilbert's Nullstellensatz
  • Theorem relating the solutions of a polynomial system to the ideal generated by the polynomials
  • States that a polynomial system has no solutions if and only if the Gröbner basis of the ideal is {1}
  • Provides a computational method to determine if a polynomial system has no solutions

Geometry of polynomial solutions

• Affine varieties
  • Set of points in affine space ($\mathbb{A}^n$) that satisfy a system of polynomial equations
  • Each solution to the polynomial system corresponds to a point on the affine variety
  • Example: the affine variety defined by $x^2 + y^2 = 1$ is a circle in the plane
• Dimension of an affine variety
  • Maximum number of independent parameters needed to describe the variety
  • Determined by the structure of the Gröbner basis (the number of free variables)
  • Example: a curve has dimension 1, a surface has dimension 2, and a hypersurface has codimension 1
• Intersection of affine varieties
  • Set of points that satisfy multiple polynomial systems simultaneously
  • Corresponds to the union of the ideals generated by the polynomial systems
  • Computed by finding the Gröbner basis of the sum of the ideals

Types of polynomial system solutions

• No solutions (inconsistent system)
  • Gröbner basis is equal to {1}, indicating that the polynomial equations have no common solutions
  • Occurs when the system is overdetermined or the equations are incompatible
  • Example: the system $x + y = 1$, $x + y = 2$ has no solutions
• Infinitely many solutions
  • Gröbner basis contains fewer univariate polynomials than the number of variables
  • System has infinitely many solutions, parametrized by the remaining free variables
  • Example: the system $x - y = 0$ has infinitely many solutions of the form $(t, t)$ for any $t$
• Overdetermined and underdetermined systems
  • Overdetermined: more equations than variables, likely to have no solutions (unless the equations are linearly dependent)
  • Underdetermined: fewer equations than variables, likely to have infinitely many solutions (parametrized by the free variables)
  • Gröbner bases can help identify and analyze these cases by revealing the structure of the solution set